You might want to start by looking at the GPS logger example sketches
. Both the Geiger counter and the GPS modules send periodic data over a serial connection. Logging Geiger counter data will therefore be a similar operation, and a similar-looking sketch.
The Geiger counter serial data lines are at 3V, so you will need a level-shifter to get them to the 5V levels the Arduino uses. You could use this level shifter chip
, or this level shifter breakout board
. You should be able to install either on the logger shield prototyping area.
Then, it's just a matter of connecting the Geiger counter serial data lines through the level shifter to a pair of Arduino digital pins (e.g., 2 and 3, as in the GPS sketch).